Eclipse Not Starting After Ubuntu 16.04 LTS Upgrade – Troubleshooting Guide

I just upgraded to 16.04 from 14.04 LTS, and suddenly Eclipse won't start. Here's the Eclipse log file:

!SESSION 2016-08-28 19:43:46.941 -----------------------------------------------
eclipse.buildId=debbuild
java.version=1.8.0_91
java.vendor=Oracle Corporation
BootLoader constants: OS=linux, ARCH=x86_64, WS=gtk, NL=en_US
Command-line arguments:  -os linux -ws gtk -arch x86_64

!ENTRY org.eclipse.osgi 2 0 2016-08-28 19:43:47.899
!MESSAGE One or more bundles are not resolved because the following root constraints are not resolved:
!SUBENTRY 1 org.eclipse.osgi 2 0 2016-08-28 19:43:47.899
!MESSAGE Bundle reference:file:plugins/org.eclipse.compare.core_3.5.200.dist.jar was not resolved.
!SUBENTRY 2 org.eclipse.compare.core 2 0 2016-08-28 19:43:47.899
!MESSAGE Missing imported package com.ibm.icu.text_3.6.1.
!SUBENTRY 2 org.eclipse.compare.core 2 0 2016-08-28 19:43:47.899
!MESSAGE Missing imported package com.ibm.icu.util_3.6.1.
!SUBENTRY 1 org.eclipse.osgi 2 0 2016-08-28 19:43:47.899
!MESSAGE Bundle reference:file:plugins/org.eclipse.ui.workbench_3.8.1.dist.jar was not resolved.
!SUBENTRY 2 org.eclipse.ui.workbench 2 0 2016-08-28 19:43:47.899
!MESSAGE Missing required capability Require-Capability: osgi.ee; filter="(|(&(osgi.ee=JavaSE)(version=1.4))(&(osgi.ee=CDC/Foundation)(version=1.1)))".
!SUBENTRY 1 org.eclipse.osgi 2 0 2016-08-28 19:43:47.900

… several pages of similar log error messages later …

!ENTRY org.eclipse.osgi 4 0 2016-08-28 19:43:48.011
!MESSAGE Application error
!STACK 1
java.lang.RuntimeException: Application "org.eclipse.ui.ide.workbench" could not be found in the registry. The applications available are: org.eclipse.ant.core.antRunner, org.eclipse.equinox.app.error, org.eclipse.equinox.p2.director, org.eclipse.equinox.p2.garbagecollector.application, org.eclipse.equinox.p2.publisher.InstallPublisher, org.eclipse.equinox.p2.publisher.EclipseGenerator, org.eclipse.equinox.p2.publisher.ProductPublisher, org.eclipse.equinox.p2.publisher.FeaturesAndBundlesPublisher, org.eclipse.equinox.p2.reconciler.application, org.eclipse.equinox.p2.repository.repo2runnable, org.eclipse.equinox.p2.repository.metadataverifier, org.eclipse.equinox.p2.artifact.repository.mirrorApplication, org.eclipse.equinox.p2.metadata.repository.mirrorApplication, org.eclipse.equinox.p2.updatesite.UpdateSitePublisher, org.eclipse.equinox.p2.publisher.UpdateSitePublisher, org.eclipse.equinox.p2.publisher.CategoryPublisher, org.eclipse.update.core.standaloneUpdate, org.eclipse.update.core.siteOptimizer, org.eclipse.pde.build.Build, org.eclipse.pde.junit.runtime.uitestapplication, org.eclipse.pde.junit.runtime.legacytestapplication, org.eclipse.pde.junit.runtime.coretestapplication, org.eclipse.pde.junit.runtime.coretestapplicationnonmain, org.eclipse.pde.junit.runtime.nonuithreadtestapplication, org.eclipse.jdt.core.JavaCodeFormatter, org.eclipse.jdt.core.JavaIndexer, org.eclipse.jdt.apt.core.aptBuild.
    at org.eclipse.equinox.internal.app.EclipseAppContainer.startDefaultApp(EclipseAppContainer.java:248)
    at org.eclipse.equinox.internal.app.MainApplicationLauncher.run(MainApplicationLauncher.java:29)
    at org.eclipse.core.runtime.internal.adaptor.EclipseAppLauncher.runApplication(EclipseAppLauncher.java:110)
    at org.eclipse.core.runtime.internal.adaptor.EclipseAppLauncher.start(EclipseAppLauncher.java:79)
    at org.eclipse.core.runtime.adaptor.EclipseStarter.run(EclipseStarter.java:353)
    at org.eclipse.core.runtime.adaptor.EclipseStarter.run(EclipseStarter.java:180)
    at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
    at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:62)
    at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
    at java.lang.reflect.Method.invoke(Method.java:498)
    at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:629)
    at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)
    at org.eclipse.equinox.launcher.Main.run(Main.java:1438)
    at org.eclipse.equinox.launcher.Main.main(Main.java:1414)

Anyone else encountered this?

Edit: upgrading to Eclipse 4.2 gives this error:

JVM terminated. Exit code=13
/usr/bin/java
-Xms40m
-Xmx512m
-XX:MaxPermSize=256m
-jar /home/_____/eclipse/eclipse//plugins/org.eclipse.equinox.launcher_1.3.0.v20120522-1813.jar
-os linux
-ws gtk
-arch x86
-showsplash /home/_____/eclipse/eclipse//plugins/org.eclipse.platform_4.2.0.v201206081400/splash.bmp
-launcher /home/_____/eclipse/eclipse/eclipse
-name Eclipse
--launcher.library /home/_____/eclipse/eclipse//plugins/org.eclipse.equinox.launcher.gtk.linux.x86_1.1.200.v20120522-1813/eclipse_1502.so
-startup /home/_____/eclipse/eclipse//plugins/org.eclipse.equinox.launcher_1.3.0.v20120522-1813.jar
--launcher.overrideVmargs
-exitdata 18800a
-vm /usr/bin/java
-vmargs
-Xms40m
-Xmx512m
-XX:MaxPermSize=256m
-jar /home/_____/eclipse/eclipse//plugins/org.eclipse.equinox.launcher_1.3.0.v20120522-1813.jar 

There seems to be a bunch of missing dependencies:

Java HotSpot(TM) 64-Bit Server VM warning: You have loaded library /opt/eclipse/plugins/org.eclipse.equinox.launcher.gtk.linux.x86_1.1.200.v20120913-144807/eclipse_1502.so which might have disabled stack guard. The VM will try to fix the stack guard now.
It's highly recommended that you fix the library with 'execstack -c <libfile>', or link it with '-z noexecstack'.
Gtk-Message: Failed to load module "overlay-scrollbar"
Gtk-Message: Failed to load module "unity-gtk-module"

(Eclipse:17614): Gtk-WARNING **: Unable to locate theme engine in module_path: "murrine",

I ran

sudo apt-get install unity-gtk-module-common:i386

but get this message:

Note, selecting 'unity-gtk-module-common' instead of 'unity-gtk-module-common:i386'
unity-gtk-module-common is already the newest version (0.0.0+15.04.20150118-0ubuntu2).
0 upgraded, 0 newly installed, 0 to remove and 1 not upgraded.

Trying to install overlay-scrollbar just says it's already the latest. Trying overlay-scrollbar:i386 gives this message:

Package overlay-scrollbar:i386 is not available, but is referred to by another package.
This may mean that the package is missing, has been obsoleted, or
is only available from another source

I also tried downgrading Java to 1.7 using sudo update-java-alternatives, which also doesn't remove the initial error